Subject: Quantum Integrability in Systems with Finite Number of Energy Levels

Since the discovery of quantum mechanics, from the Bohr atom and the harmonic oscillator, to the present day, quantum integrable models have played a central role in our understanding of physics at the quantum level. Recently, the field has acquired a new prominence with a range of solid state and cold atom experiments, which demonstrate that integrable systems fail to equilibrate, and thereby defy a conventional statistical description. Roughly speaking, a quantum integrable system is one whose quantum Hamiltonian contains additional integrals of motion beyond the usual total energy and momenta. Yet a complete, unambiguous notion of quantum integrability has long remained elusive, and our understanding of its nonequilibrium and other manifestations is correspondingly incomplete. In the opposite case of chaotic systems, Random Matrix Theory famously provides a tremendously successful analysis of their universal properties. In this talk, I will propose a surprisingly simple and yet unambiguous notion of quantum integrability which leads to a clear explanation and delineation of its various features, culminating in Integrable Matrix Theory: a counterpart of Random Matrix Theory for integrable quantum Hamiltonians.